ATM 01Correctly label angles. Name the same angle in all possible correct ways. Also be able to identify points on the interior, exterior, and vertex. Additional terms: congruent angles, adjacent angles, shared rays, angle bisector, opposite rays.
An angle, or two rays meeting at one endpoint, can be named in several different ways. Referring to the diagram at right, we can name this angle as follows:
Angle Q: Naming angles using the vertex (not the most specific, as some angles may share a vertex) Angle RQS: Naming angles with 3 points on the angle, the middle one being the vertex Angle 1: Naming angles with numbers There are various names for the points in the first diagram as well: Point Q: The vertex Point R: On the angle Point U: On the interior Point T: On the exterior There are names for various angles, as seen in the second diagram at left: ∠AXC and ∠CXB are congruent angles (≅) as well as adjacent angles (next to each other) Ray XC is a shared ray of ∠AXC and ∠CXB, as well as the angle bisector of ∠AXB |
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ATM 02Given an angle, measure it with a protractor. Given an angle measurement, straight edge and protractor, draw an angle with that measurement.
Angles are measured with a tool called a protractor, as seen at right. The usual unit used for measuring angles in geometry is degrees, which are 1/360 of a full circle (a straight line is 180 degrees around a full circle).
1. To measure an angle, line up the vertex with the small hole or cross-hatch in the middle of the straight side of the protractor as seen in the picture. 2. Line the straight lines on either side of the hole or cross-hatch with one of the rays and the other ray with one of the degree-marks on the curved side of the protractor (Note: if the ray is too short to line up with one of the marks on the protractor, carefully extend the ray with a pencil and the straight edge of the protractor). 3. Each longer unmarked line is 5 degrees, while each shorter line is one degree. Usually rounding to the near nearest degree, the line that matches up the best with the second ray is the measure of the angle. To draw an angle, you follow almost the same steps, but you have to draw your own ray before step 1 using a straight edge. Then you make a mark at the desired degree and connect it with a vertex using a straight edge. |
ATM 03Classify angles including the terms of Zero, Acute, Right, Obtuse, Straight or Reflex.
Referring to the diagram at left and assuming drawings are to scale, we can say the following:
1. This angle is an acute angle. Acute angles measure less than 90 degrees. 2. This angle is an obtuse angle. Obtuse angles measure over 90 degrees but less than 180 degrees. 3. This angle is a right angle. It is exactly 90 degrees. 4. This angle is a straight angle. It is exactly 180 degrees. 5. This angle (marked) is a reflex angle. it is over 180 degrees but less than 360 degrees. 6. This angle is a zero angle. It is basically two rays overlapping and is 0 degrees wide. |
ATM 04State the Angle Addition Postulate using variables, then use substitution to evaluate the angle measures for a given problem. Use correct notation.
The Angle Addition Postulate is almost exactly like the Segment Addition Postulate covered in the last unit (BFF), except with angles. Referring to the diagram at left, it can be written as follows:
m∠ABC + m∠CBD = m∠ABD Assuming that m∠ABC=35 and m∠CBD=27, what is the m∠ABD? Start with the postulate in variables: m∠ABC + m∠CBD = m∠ABD Substitute: 35 + 27 = m∠ABD Solve: 72 = m∠ABD Extra Resources Lesson on study.com |
ATM 06Given an angle, sketch an angle bisector correctly labeling the drawing. Also solve problems involving bisected angles.
To sketch an angle bisector using only a compass, follow the procedure below, referring to the diagrams at left: 1. Starting with the angle at left, place the needlepoint of the compass on B (the vertex) and draw an arc intersecting with both rays. 2. Placing the needlepoint on the point where the arc intersects with each ray, set the compass more than halfway across the angle and draw a small arc (Note: The top point where the arcs touch is the only point you need, so the minimum amount of arc for each may look vaguely like an X on the interior of the angle). 3. Using a straight edge, connect the point where the previous two arcs drawn connected with the vertex. Make the bisector a ray, because that is the only object that can bisect angles. 4. While the bisector has been drawn, the diagram is not complete without marking that the ray is a bisector and that the two angles created are congruent. Either draw the same number of arcs in each angle to show congruence or draw one arc with the same number of slashes through them. See the last diagram. Extra Resources
Explanation by geometryvids |
ATM 07 State the Angle Overlap Theorem (include a sketch of the corresponding diagram) using variables, then apply the theorem to find measures of angles. SHOW WORK.
The Angle Overlap Theorem follows the same general structure as the Segment Overlap Theorem. Using the diagram at right, we can state AOT in two ways:
If m∠AXC=m∠BXD, then m∠AXB=m∠CXD We can use this to solve for different angle measures. For example: m∠AXC=38 and m∠AXB=50 Find m∠CXB Solving using substitution, we get: m∠AXB - m∠AXC = m∠CXB (50) - (38) = m∠CXB 12 = m∠CXB |
Explanation by Mr. Erlin, 2015
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ATM 08Derive, using Algebra, the Angle Overlap Theorem (include a sketch of the corresponding diagram) using variables.
This is another two-column proof. It is very similar to deriving the Segment Overlap Theorem, but with the Angle Addition Postulate Substituting for the Angle Addition Postulate. Refer to the diagram and statement in the previous problem for the proof.
QED
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ATM 50 Construct two congruent angles (using compass…NOT protractor).
To copy an angle, follow the procedure below and refer to the diagrams at left: 1. Draw a ray that is obviously longer than the first ray of the angle you want to copy. 2. Setting the compass's needlepoint on the vertex and opening it to any distance, draw an arc that intersects both rays. 3. Keeping the compass open the same distance, set the needlepoint on the vertex of the copied angle and draw a similarly-wide arc. 4. Set the needlepoint where the arc intersects with the first ray on the original angle and open it to where the arc intersects with the second ray. Make a small construction mark. 5. Repeat Step 4 on the copied angle, but keep the compass open the same distance when drawing a construction mark on the second angle. 6. Connect the vertex to where the arc and the construction mark intersect using a straight edge. 7. Use multiple arcs or arcs with slashes through them to mark that the angle was copied exactly. |
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ATM 51Construct an angle that is the sum of two given angles.
This construction is similar to copying two angles so that they share a side, then taking out that shared side. Watch the video to the left for clarification.
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ATM 52Construct an angle bisector of a given angle.
To construct an angle bisector, follow these steps:
1. Set point of compass on vertex of angle; open to any width 2. Draw an arc that intersects both sides of the angle 3. Using the two points of intersection you just made as your vertex, draw two intersecting arcs the same distance from their respective vertices (the curved x in the diagram) 4. Connect the new point of intersection with the original angle's vertex using a straight edge (it should look like the diagram to the left) |
Images courtesy of USF
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ATM 99Solve problems you haven’t seen before, using analysis and synthesis of the information learned so far.
Find the number of degrees between the hands of a clock at 4:08. Then find the number of degrees between the hands of a clock at 4:32, and explain how to do this also. Are there any differences in the method used for 4:08 and 4:32? Can you write an equation? Hint: How many degrees are there between each minute mark? Be sure to remember to include the movement of the hour hand as well. Extra Resources Clock Problems on mathforum.org |